Nef cones of nested Hilbert schemes of points on surfaces

Abstract

Let $X$ be the projective plane, a Hirzebruch surface, or a general $K3$ surface. In this paper, we study the birational geometry of various nested Hilbert schemes of points parameterizing pairs of zero-dimensional subschemes on $X$. We calculate the nef cone for two types of nested Hilbert schemes: $X^{[n+1,n]}$ and universal families. As an application, we recover a theorem of Butler on syzygies on Hirzebruch surfaces.